Spit Chain: Social Network Analysis in Nonhuman Primates
Sam, Nicole, Jess, Bhavya, Brooke
2023-10-29
What is a social network?
A social network analysis describes all the individuals in a group AND all the links between them. The photo above shows a social network from the show The L Word. These are all individuals who have romantic and sexual connections to each other.
Terms to know
Nodes: the individuals in the social network
Edges: the relationships between individuals in a network
Directionality: some social networks will have a relationship that goes in one direction (e.g., Bonobo A grooms Bonobo B. Other relationships won’t have directionality, such as friendship). Directionality is relevant for understanding transactional or rank-based relationships.
Edge weight: the strength of the relationship between individuals, which may differ between individuals within social networks (e.g., if A gives B money, we might be interested the amount they gave. This value is the edge’s weight)
By the end of this module, you should be able to answer the following
questions about social networks in R:
1) How are individuals
connected in a group?
2) How can we test the extent to
which specific attributes influence whether individuals in a group are
connected?
3) How do networks change over time?
To accomplish this, we will use social network datasets from nonhuman primates! We will start with a dataset from bonobos. Bonobos are one of our closest living relatives and live in multi-male/multi-female social groups. Female bonobos are highly cohesive and have strong affiliative bonds. These bonds are maintained through high levels of sociosexual behavior, the most common being genito-genital (GG) rubbing. But similar to other primates, bonobos will also groom each other.
Installing packages
Here are the packages we will be using:
library(statnet)
library(igraph)
library(intergraph){igraph} and {statnet} are two of the most commonly used packages for social network analyses. Both include built-in functions that can plot networks and answer almost every beginner-level question about a given network. However, there may be some functions that are present in one package but not the other (or just work better). So, {intergraph} is another useful package that helps users transition between {igraph} and {statnet}.
Understanding the structure of social network data
Loading data
Let’s start by loading our data from bonobos. There are two datasets from the same bonobo population. One describes their GG rubbing network and the other describes their grooming networks.
library(curl)
f <- curl("https://raw.githubusercontent.com/nmerullo/AN588_BonoboSpitChain/main/AnzaEtAl_2021_ggr_weighted.csv")
bonobo_ggr_edgelist <- read.csv(f, header = TRUE, sep = ",")
head(bonobo_ggr_edgelist) # data about bonobos' GG rubbing interactions## actor receiver GGR.Weight
## 1 ln ky 4
## 2 h j 20
## 3 j h 4
## 4 ky lc 15
## 5 lc ky 20
## 6 lc ln 31q <- curl("https://raw.githubusercontent.com/nmerullo/AN588_BonoboSpitChain/main/AnzaEtAl_2021_grooming_weighted.csv")
bonobo_groom_edgelist <- read.csv(q, header = TRUE, sep = ",")
head(bonobo_groom_edgelist) # data about bonobos' grooming interactions## actor receiver weight
## 1 j li 1
## 2 li h 3
## 3 h j 6
## 4 j h 3
## 5 h li 2
## 6 ln dn 8What are we looking at in these data sets? Who are the actors (also known as senders) vs the receivers? In the first 6 rows we are looking at for each data set, who has the most weighted relationship? What does this mean?
Setting up social network data
How we set-up and manage the social network data can impact the ease with which we navigate our social network analysis. Based on how many individuals there are, whether we know any of their attributes (age, sex, etc), and information about the edges themselves (directionality, edge-weights, etc), we can store and input data either as a matrix or an edgelist.
Adjacency matrix
A matrix used for social network analysis is called a
sociomatrix or an adjacency matrix.
They are square matrices. For directed networks, rows indicate the
starting node, and columns indicate the receiving node. Undirected
networks are symmetric matrices around the diagonal. For un-weighted
networks, the cells contain a 1, for the presence of an edge and a 0 for
the absence of an edge between individuals. For weighted networks, the
cells can contain the edge-weights instead of 1s and 0s. 
Edge list
Alternatively, we can also represent network data as a columned list, called an edge list. Each row of the list represents one edge (one relationship between individuals). In directed data, the first column names the sender of the edge and the second column names the receiver. In undirected data, it does not matter which node is listed first. For weighted data, a third column contains a numeric value for each tie.
Is our bonobo dataset structured as an edge list or adjacency matrix?
Node-Level Variables
To include variables that describe your nodes (age, rank, sex, etc), it is good practice to load in a separate file. This data frame must include one column of the node IDs using the exact same IDs used in the adjacency matrix or edge list. All other variables should then be included in additional columns.
j <- curl("https://raw.githubusercontent.com/nmerullo/AN588_BonoboSpitChain/main/AnzaEtAl_2021_attributes.csv")
bonobo_attribute <- read.csv(j, header = TRUE, sep = ",")
bonobo_attribute## actor rank age
## 1 ln 3.42 9
## 2 h 3.42 31
## 3 j 6.20 24
## 4 ky 4.66 11
## 5 lc 4.11 9
## 6 dn 5.57 44
## 7 li 5.23 12Who is the highest ranking bonobo? Who is the lowest? Who is the oldest? Who is the youngest?
Now we can create the network objects and add node attributes to the objects. Note that adding node attributes to objects involves ‘%v%’, which resembles a pipe. This functions similarly to a $ operator and allows us to add rank variables to a dataset.
bonobo_ggr_net <- as.network.data.frame(bonobo_ggr_edgelist) # turns GG rubbing edge list into a GG rubbing network
bonobo_groom_net<-as.network.data.frame(bonobo_groom_edgelist) # turns grooming edge list into a grooming network
bonobo_ggr_net %v% "rank" <- bonobo_attribute$rank #adds rank variable to GG rubbing network as a vertex attribute
bonobo_ggr_net %v% "age" <- bonobo_attribute$age #adds age variable to GG rubbing network as a vertex attribute
bonobo_groom_net %v% "rank" <- bonobo_attribute$rank #adds rank variable to grooming network as a vertex attribute
bonobo_groom_net %v% "age" <- bonobo_attribute$age #adds age variable to grooming network as a vertex attributeNow that everything it set up, we can use summary() to look at our networks. For example, let’s take a look at the bonobo grooming network.
summary(bonobo_ggr_net)## Network attributes:
## vertices = 7
## directed = TRUE
## hyper = FALSE
## loops = FALSE
## multiple = FALSE
## bipartite = FALSE
## total edges = 13
## missing edges = 0
## non-missing edges = 13
## density = 0.3095238
##
## Vertex attributes:
##
## age:
## integer valued attribute
## 7 values
##
## rank:
## numeric valued attribute
## attribute summary:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 3.420 3.765 4.660 4.659 5.400 6.200
## vertex.names:
## character valued attribute
## 7 valid vertex names
##
## Edge attributes:
##
## GGR.Weight:
## integer valued attribute
## 13values
##
## Network adjacency matrix:
## ln h j ky lc li dn
## ln 0 0 0 1 1 0 1
## h 0 0 1 0 0 1 0
## j 0 1 0 0 0 0 0
## ky 1 0 0 0 1 0 0
## lc 1 0 0 1 0 0 1
## li 0 1 0 0 0 0 0
## dn 0 0 0 0 1 0 0Using what we have learned so far, how can we interpret the output of this summary?
Visualizing social network data
We can visualize social networks by creating network objects with {statnet}. Note the arrows indicating directionality!
par(mfrow = c(1, 2))
plot(bonobo_ggr_net, label = "vertex.names", main="GG rubbing")
box(col = "black")
plot(bonobo_groom_net, label = "vertex.names", main="Grooming")
box(col = "black")
How would you describe these two plots? What stands out? What’s different?
Quadratic Assignment Procedure (QAP)
When analyzing social network data, we want to know to what degree our variables impact the formation of a particular network organization. Our independent variables or predictor variables are node level variables (e.g. age) or similarity between node level variables (e.g. whether pairs belong to the same age category). Our dependent variable is the organization of edges in the network.
We cannot use standard OLS regressions to analyze pair characteristics and the presence of edges between pairs because it violates the assumption that observations are independent. For example, observations A-B, A-C, and A-D are not independent because they all involve individual A. Or if one bonobo is GG rubbing with 5 other bonobos, those 5 other bonobos may also be GG rubbing with each other. Quadratic assignment procedure (QAP) addresses this issue of non-independence.
QAP is a resampling-based method that controls for non-independence
in network structure via random permutations. These permutations use
different arrangements of the rows and columns in the adjacency matrix.
Thus, the network structure is maintained but the arrangement of
individuals in the structure is randomized. This represents the null
hypothesis because it should eliminate any potential correlations
between ties and independent values. 
What kinds of questions would you ask with our bonobo data?
There are 3 types of QAP regressions we can run with {statnet}: correlation regressions, multiple/linear regressions, and logistic regressions. We will describe and demonstrate each regression using the bonobo data. We’ll also do the same with a more complex dataset from spider monkeys!
1. Correlation
Correlation QAPs simply test whether two social networks are correlated. For example, using our bonobo datasets, we can ask: Do bonobos who groom each other correlate with bonobos who GG rub with one another?
What is our null hypothesis and alternative hypothesis in this question?
We can start by using the gcor() function to get a correlation coefficient between the two networks.
gcor(bonobo_ggr_net, bonobo_groom_net)## [1] 0.1111054What does this correlation coefficient signify?
These networks don’t seem very correlated but let’s test for significance anyways. The qaptest() function uses Monte Carlo simulations of likelihood quantiles to test arbitrary graph-level statistics against a QAP null hypothesis.
qap_cor <- qaptest(list(bonobo_ggr_net, bonobo_groom_net), # include both network objects in a list
gcor, # the function you're using is correlation between networks (gcor)
g1=1, # use first graph in the list (in this case the GG rubbing network)
g2=2, # use second graph in the list (in this case the groom network)
reps = 1000) # number of permutations to run
summary(qap_cor)##
## QAP Test Results
##
## Estimated p-values:
## p(f(perm) >= f(d)): 0.317
## p(f(perm) <= f(d)): 0.837
##
## Test Diagnostics:
## Test Value (f(d)): 0.1111054
## Replications: 1000
## Distribution Summary:
## Min: -0.4191705
## 1stQ: -0.1275187
## Med: 0.005050247
## Mean: -0.01361547
## 3rdQ: 0.1111054
## Max: 0.8534918Under “Test Diagnostics”, the test value represents the correlation coefficient between grooming and GG rubbing networks. This is interpreted at the level of the dyad, pairs who groom each other are likely not related to pairs who GG rub with one another. We previously calculated this using the gcor() function.
Under “Estimated p-values”, we find statistics that tell us about how this observed correlation compares to what the correlation looks like when we randomly permute one of our networks. Note that these are both one-tailed p-values! We can see that our p-values are greater than 0.05, meaning we have evidence to fail to reject our null hypothesis. There is no significant correlation between grooming and GG rubbing networks.
We can also visualize the “Distribution Summary” using the plot() function.
plot(qap_cor)
The dotted line gives us the correlation among the observed networks. The curve shows the distribution of correlations across permutated networks. Our plot shows that across 1000 replications, very rarely do we see any correlation between grooming and GG rubbing networks in this bonobo population. The correlation tends to center around 0.
2. Linear/Multiple regression
Linear regression QAPs test the extent to which predictor variables are affecting edge weights. Recall that our predictor variables are node attributes.
For example, if our predictor variable is rank, we can ask: Does rank influence the edge weight of GG rubbing? In other words, we are asking: Is a bonobo more likely to have more weighted edges for each unit increase in rank?
Before we dive in, what do you predict? Do you think there is be a difference between actors (senders) and receivers? What does it mean to have more edge weight?
Predictor variables must be matrices in order to be used in the QAP functions. We can look at whether rank affects the weight of ties sent or received (or both, but they must be added to the model separately).
First, we must create a matrix that shows the rank of senders. Note that senders are represented by ROWS in the matrices.
nodes <- length(bonobo_attribute$actor) # number of nodes in the data set
rank <- bonobo_ggr_net %v% "rank" # a vector of the node-level variable we are interested in
rank_sending <- matrix(data = NA, nrow = nodes, ncol = nodes) # create empty matrix to be filled for senders
for (i in 1:nodes){ # for 1 through the number of nodes in the data set
rank_sending[i,] <- rep(rank[i], nodes)} # the rank of each actor is repeated over entire ROW of matrix
rank_sending## [,1] [,2] [,3] [,4] [,5] [,6] [,7]
## [1,] 3.42 3.42 3.42 3.42 3.42 3.42 3.42
## [2,] 3.42 3.42 3.42 3.42 3.42 3.42 3.42
## [3,] 6.20 6.20 6.20 6.20 6.20 6.20 6.20
## [4,] 4.66 4.66 4.66 4.66 4.66 4.66 4.66
## [5,] 4.11 4.11 4.11 4.11 4.11 4.11 4.11
## [6,] 5.57 5.57 5.57 5.57 5.57 5.57 5.57
## [7,] 5.23 5.23 5.23 5.23 5.23 5.23 5.23Next, we must create a matrix that shows the rank of receivers. Note that receivers are represented by COLUMNS in the matrices.
rank_receiving <- matrix(data = NA, nrow = nodes, ncol = nodes) # empty matrix for receivers
for (i in 1:nodes){ # for 1 through the number of nodes in the data set
rank_receiving[,i] <- rep(rank[i], nodes)} # the rank of each receiver is repeated over entire COLUMN of matrix
rank_receiving## [,1] [,2] [,3] [,4] [,5] [,6] [,7]
## [1,] 3.42 3.42 6.2 4.66 4.11 5.57 5.23
## [2,] 3.42 3.42 6.2 4.66 4.11 5.57 5.23
## [3,] 3.42 3.42 6.2 4.66 4.11 5.57 5.23
## [4,] 3.42 3.42 6.2 4.66 4.11 5.57 5.23
## [5,] 3.42 3.42 6.2 4.66 4.11 5.57 5.23
## [6,] 3.42 3.42 6.2 4.66 4.11 5.57 5.23
## [7,] 3.42 3.42 6.2 4.66 4.11 5.57 5.23Now we can run our linear regression! First we are asking: Does rank affect how many times bonobos GG rubbed others (the weight of the edges they sent)? Null hypothesis is that rank does not affect how many times bonobods GG rub others (both high and low)
lrqap <- netlm(bonobo_ggr_net, # response variable is a network object with a weighted adjacency matrix
list(rank_sending)) # list of predictor variables as network or matrix objects
summary(lrqap)##
## OLS Network Model
##
## Residuals:
## 0% 25% 50% 75% 100%
## -0.4534205 -0.3093578 -0.2036344 0.5465795 0.8695587
##
## Coefficients:
## Estimate Pr(<=b) Pr(>=b) Pr(>=|b|)
## (intercept) 0.8507548 0.998 0.002 0.004
## x1 -0.1161796 0.008 0.992 0.027
##
## Residual standard error: 0.4587 on 40 degrees of freedom
## Multiple R-squared: 0.06227 Adjusted R-squared: 0.03883
## F-statistic: 2.656 on 1 and 40 degrees of freedom, p-value: 0.111
##
##
## Test Diagnostics:
##
## Null Hypothesis: qap
## Replications: 1000
## Coefficient Distribution Summary:
##
## (intercept) x1
## Min -2.660544 -1.882904
## 1stQ -0.654187 -0.620122
## Median 0.091024 -0.007029
## Mean 0.077896 0.004873
## 3rdQ 0.849541 0.522875
## Max 2.693891 1.874392To interpret this we can imagine the x axis being rank and the y axis being edge weight. For every unit increase in rank, edge weights are going down 0.1161796 (the estimate of x1 in the above coefficients). This relationship is slightly significant at 0.032 on the two tailed test. However, if we look at our adjusted r^2 value, our model is only explaining ~4% of the variability. And our F-statistic p value is too high to be significant at 0.111. Taken together, we can say that the rank of sender is not significantly explaining the edge weight of GG rubbing interactions.
Now let’s look at the receivers.
Does rank affect how many times bonobos received GG rubbing from others (the weight of the edges they receive)? The null hypothesis is the same as above, that rank of receiver does not affect the tnumber of times they are GG rubbed with.
lrqap2 <- netlm(bonobo_ggr_net, # response variable is a network object with a weighted adjacency matrix
list(rank_receiving)) # list of predictor variables as network or matrix objects
summary(lrqap2)##
## OLS Network Model
##
## Residuals:
## 0% 25% 50% 75% 100%
## -0.3992884 -0.3393159 -0.2434687 0.6007116 0.8021901
##
## Coefficients:
## Estimate Pr(<=b) Pr(>=b) Pr(>=|b|)
## (intercept) 0.64715036 0.993 0.007 0.012
## x1 -0.07247427 0.047 0.953 0.103
##
## Residual standard error: 0.4679 on 40 degrees of freedom
## Multiple R-squared: 0.02423 Adjusted R-squared: -0.0001605
## F-statistic: 0.9934 on 1 and 40 degrees of freedom, p-value: 0.3249
##
##
## Test Diagnostics:
##
## Null Hypothesis: qap
## Replications: 1000
## Coefficient Distribution Summary:
##
## (intercept) x1
## Min -2.11668 -1.23372
## 1stQ -0.56714 -0.48077
## Median 0.04401 0.03890
## Mean 0.04428 0.01741
## 3rdQ 0.66126 0.51956
## Max 2.00229 1.34781Just like with the actors/senders, let’s imagine that rank is on the x axis and edge weight is on the y axis. Again, we see a negative slope, though this one is even less steep at -0.07247427. The two tailed test p value for this variable is not significant. This model is explaining almost 0% of the variability, which means this model is really not a good fit for the residuals. We can imagine the points scattered far away from the line. The F-statistic p-value is too high to be significant at 0.3249, which is even higher than the sender p-value. Rank of receiver does not significantly explain the edge weight of GG rubbing interactions.
So we fail to reject the null hypothesis in this question too. But that begs the question, if rank is not influencing the edge weight then is there some other attribute that is?
We can include multiple predictor variables! We can also use another network as a predictor variable! By doing so, we are conducting a multiple regression QAP. For example, we can ask: Do sender rank and grooming relationships predict the number of times bonobos GG rubbed in the network? The null hypothesis is that neither rank of of the actor nor grooming relationships affect the number of times bonobos initiate GG rubbing (two tailed test).
mrqap <- netlm(bonobo_ggr_net, # response variable is a network object with a weighted adjacency matrix
list(rank_sending, bonobo_groom_net)) # list of all predictor variables as network or matrix objects
summary(mrqap)##
## OLS Network Model
##
## Residuals:
## 0% 25% 50% 75% 100%
## -0.5370306 -0.3281637 -0.1926087 0.5464810 0.8485420
##
## Coefficients:
## Estimate Pr(<=b) Pr(>=b) Pr(>=|b|)
## (intercept) 0.8256023 0.996 0.004 0.008
## x1 -0.1210313 0.007 0.993 0.021
## x2 0.1253554 0.790 0.210 0.484
##
## Residual standard error: 0.4603 on 39 degrees of freedom
## Multiple R-squared: 0.07951 Adjusted R-squared: 0.0323
## F-statistic: 1.684 on 2 and 39 degrees of freedom, p-value: 0.1988
##
##
## Test Diagnostics:
##
## Null Hypothesis: qap
## Replications: 1000
## Coefficient Distribution Summary:
##
## (intercept) x1 x2
## Min -2.555843 -2.073998 -2.949316
## 1stQ -0.610343 -0.574280 -0.932728
## Median 0.104895 -0.017388 -0.143389
## Mean 0.085593 0.005107 0.095635
## 3rdQ 0.830378 0.507682 0.725760
## Max 2.608258 1.985326 10.532012Are any of the above relationships significant when taken into the model together? Is this model a better fit than the others? How can we tell?
3. Logistic regression
Logistic regression QAPs test the extent to which independent
variables are affecting the presence or absence of edges, but not the
weight of those edges. If you have a network with binary ties, we can
use this regression method to predict ties in the outcome network
(response variable)
Using our bonobo dataset, we can ask:
Does older age make an individual more or less likely to G-G rub
with other individuals?
In this example, our GG rubbing
network is the response variable while age is our predictor
variable.
If you imagine age (as the predictor variable) on the x axis, for every year a bonobo gets older, are they more likely to GG rub with more individuals? Will an older individual have more ties on the social network?
Similar to the multiple regression, we must first create a matrix that shows the rank of senders and receivers. As a reminder, senders are represented by ROWS while receivers are represented by COLUMNS in matrices
nodes <- length(bonobo_attribute$actor) # number of nodes in the data set
age <- bonobo_ggr_net %v% "rank" # a vector of the node-level variable we're interested in
age_sending <- matrix(data = NA, nrow = nodes, ncol = nodes) #create empty matrix to be filled
age_receiving <- matrix(data = NA, nrow = nodes, ncol = nodes)
for (i in 1:nodes){ # for 1 through the number of nodes in the data set
age_sending[i,] <- rep(age[i], nodes)} # The age of each actor is repeated over entire ROW of matrix
age_sending## [,1] [,2] [,3] [,4] [,5] [,6] [,7]
## [1,] 3.42 3.42 3.42 3.42 3.42 3.42 3.42
## [2,] 3.42 3.42 3.42 3.42 3.42 3.42 3.42
## [3,] 6.20 6.20 6.20 6.20 6.20 6.20 6.20
## [4,] 4.66 4.66 4.66 4.66 4.66 4.66 4.66
## [5,] 4.11 4.11 4.11 4.11 4.11 4.11 4.11
## [6,] 5.57 5.57 5.57 5.57 5.57 5.57 5.57
## [7,] 5.23 5.23 5.23 5.23 5.23 5.23 5.23for (i in 1:nodes){
age_receiving[,i] <- rep(age[i], nodes)}
age_receiving## [,1] [,2] [,3] [,4] [,5] [,6] [,7]
## [1,] 3.42 3.42 6.2 4.66 4.11 5.57 5.23
## [2,] 3.42 3.42 6.2 4.66 4.11 5.57 5.23
## [3,] 3.42 3.42 6.2 4.66 4.11 5.57 5.23
## [4,] 3.42 3.42 6.2 4.66 4.11 5.57 5.23
## [5,] 3.42 3.42 6.2 4.66 4.11 5.57 5.23
## [6,] 3.42 3.42 6.2 4.66 4.11 5.57 5.23
## [7,] 3.42 3.42 6.2 4.66 4.11 5.57 5.23Now we can run our logistic regression!
What is our null hypothesis for this question?
The netlogit() function performs a logistic regression of the network variable for the response variable (bonobo_ggr_net) on the network variables in the predictor variables (age_receiving and age_sending). The resulting fits and coefficients are tested against the null hypothesis.
logqap <- netlogit(bonobo_ggr_net, # response variable is a network object with an unweighted adjacency matrix
list(age_receiving, age_sending), # list of all predictor variables as network or matrix objects
reps = 1000) # number of draws for quantile estimation, 1000 reps is the default
summary(logqap)##
## Network Logit Model
##
## Coefficients:
## Estimate Exp(b) Pr(<=b) Pr(>=b) Pr(>=|b|)
## (intercept) 4.5524763 94.8670398 0.973 0.027 0.054
## x1 -0.4903676 0.6124013 0.033 0.967 0.065
## x2 -0.6817886 0.5057117 0.018 0.982 0.024
##
## Goodness of Fit Statistics:
##
## Null deviance: 58.22436 on 42 degrees of freedom
## Residual deviance: 47.45396 on 39 degrees of freedom
## Chi-Squared test of fit improvement:
## 10.77041 on 3 degrees of freedom, p-value 0.01303442
## AIC: 53.45396 BIC: 58.66696
## Pseudo-R^2 Measures:
## (Dn-Dr)/(Dn-Dr+dfn): 0.2040994
## (Dn-Dr)/Dn: 0.1849811
## Contingency Table (predicted (rows) x actual (cols)):
##
## Actual
## Predicted 0 1
## 0 25 11
## 1 4 2
##
## Total Fraction Correct: 0.6428571
## Fraction Predicted 1s Correct: 0.3333333
## Fraction Predicted 0s Correct: 0.6944444
## False Negative Rate: 0.8461538
## False Positive Rate: 0.137931
##
## Test Diagnostics:
##
## Null Hypothesis: qap
## Replications: 1000
## Distribution Summary:
##
## (intercept) x1 x2
## Min -2.04329 -1.57848 -2.01970
## 1stQ -0.73472 -0.53225 -0.63740
## Median -0.05205 0.03693 -0.05645
## Mean -0.03918 0.02587 -0.04178
## 3rdQ 0.63445 0.69756 0.55504
## Max 2.04528 1.63277 1.99509As a reminder, we are testing whether age affects the likelihood of giving and receiving GG rubbing. In the output above, x1 represents the first variable in the model (age_receiving), and x2 represents the second variable in the model (age_sending).
There is a significantly negative relationship between age of sender and likelihood of number of edges in the GG rubbing network. In other words, for every year older, a female bonobo has -0.6817886 fewer GG rubbing edges for initiating. They are less likely to initiate GG rubbing as they get older. However, we cannot say anything about getting older and receiving GG rubbing events because the relationship was not significant for the two tailed hypothesis. Though we can note the relationship is also negative for this. The Chi-squared goodness of fit test p-value is signficant.
A follow up question given these results could be are older bonobos overall less likely to participate in GG rubbing? We could answer this with a linear regression QAP.
Our bonobo data was a little anti-climactic. But sometimes, that’s the nature of small sample primatology work. To make things a little more exciting, let’s run the three QAPs again with some spider monkey data.
Spider Monkey Social Networks
The spider monkeys in this data set are Ateles hybridus or
brown spider monkeys. They are a group living primate with
fission-fusion dynamics, similar to bonobos. Spider monkeys, however are
a new world monkey. The original intent of this data set and what it was
published for was to test how parasites moved between individuals
depending on different levels of contact. That leaves us with a rich
dataset of networks and attributes.
First, we must load our data.
spmonkey_dat <- curl("https://raw.githubusercontent.com/nmerullo/AN588_BonoboSpitChain/main/spidermonkey_beh_edgelist.csv")
spmonkey_el <- read.csv(spmonkey_dat)
head(spmonkey_el)## Focal Beh Rec Mins
## 1 Ba In contact Bo 262
## 2 Ba In contact Db 1
## 3 Ba In contact Dl 134
## 4 Ba In contact Ku 6
## 5 Ba In contact Nw 10
## 6 Ba In contact Pe 37unique(spmonkey_el$Beh) #subsetting the column for proximity types## [1] "In contact" "mating" "play" "play " "groom"We see here that spider monkeys are engaging in four unique social behaviors in this dataset: in contact, mating, playing, and grooming. This means that we can construct four separate social networks. Additionally, the packages that we use can only read edge lists that are in the form of a 2-column format. So, let’s clean this data up a bit more.
First, we are starting with the mating network.
mating_spmonkey <- spmonkey_el[spmonkey_el$Beh == "mating",] #making a separate data frame for mating
mating_spmonkey <- mating_spmonkey[,-c(2,4)] #removing unnecessary column data, we'll add back this "Mins" data as edge attributes later
sp_mating_net <- as.network(mating_spmonkey, matrix.type = "edgelist", directed = TRUE)
plot(sp_mating_net, label = "vertex.names", main="Spider Monkey Mating Network")
Next, let’s add network and edge attributes to the network.
sp_attr <- curl("https://raw.githubusercontent.com/nmerullo/AN588_BonoboSpitChain/main/spidermonkey_attributes.csv")
sp_attr <- read.csv(sp_attr)
head(sp_attr)## Number ID sex age dead.alive Repr_status
## 1 1 Pe female adult dead preg/lact
## 2 2 Ku female adult dead preg/lact
## 3 3 Dl female adult dead preg/lact
## 4 4 Ba female adult dead preg/lact
## 5 5 Vi female adult dead preg/lact
## 6 6 Nw male adult alive repr_activesp_attr$age[sp_attr$age == "infant"] <- 1 #assigning a numeric value to each life history stage
sp_attr$age[sp_attr$age == "juvenile"] <- 2
sp_attr$age[sp_attr$age == "adult"] <- 3
sp_attr$age <- as.numeric(sp_attr$age)
class(sp_attr$age) #confirming the age attribute is coming up as a numeric class## [1] "numeric"sp_attr$sex[sp_attr$sex == "female"] <- 0 #assigning a numeric value to each sex
sp_attr$sex[sp_attr$sex == "male"] <- 1
sp_attr$sex <- as.numeric(sp_attr$sex)
class(sp_attr$sex) #confirming the sex attribute is coming up as a numeric class## [1] "numeric"#next we need to assign the node attributes to the mating network
sp_mating_net %v% "sex" <- sp_attr$sex #adds sex as node attribute
sp_mating_net %v% "age" <- sp_attr$age #adds age as node attribute
sp_mating_net %v% "repr" <- sp_attr$Repr_status #adds reproductive status as a node attributeWe can also add the “Mins” from the initial spider monkey edge list as edge weights. The %e% operator is like the %v% operator. It subsets a network object to the edge attribute so we can store time (in minutes spent mating) there.
sp_mating_net %e% "time" <- spmonkey_el[spmonkey_el$Beh == "mating",]$Mins
summary(sp_mating_net) #can summarize the network like we did for the bonobo GG rubbing network## Network attributes:
## vertices = 7
## directed = TRUE
## hyper = FALSE
## loops = FALSE
## multiple = FALSE
## bipartite = FALSE
## total edges = 11
## missing edges = 0
## non-missing edges = 11
## density = 0.2619048
##
## Vertex attributes:
##
## age:
## numeric valued attribute
## attribute summary:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 3 3 3 3 3 3
##
## repr:
## character valued attribute
## attribute summary:
## preg/lact repr_active
## 5 2
##
## sex:
## numeric valued attribute
## attribute summary:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.0000 0.0000 0.2857 0.5000 1.0000
## vertex.names:
## character valued attribute
## 7 valid vertex names
##
## Edge attributes:
##
## time:
## integer valued attribute
## 11values
##
## Network adjacency matrix:
## Nw Ba Pe Wa Dl Pk Vt
## Nw 0 1 1 0 1 0 0
## Ba 1 0 0 1 0 1 1
## Pe 1 0 0 1 0 1 0
## Wa 0 0 0 0 1 0 0
## Dl 0 0 0 0 0 0 0
## Pk 0 0 0 0 0 0 0
## Vt 0 0 0 0 0 0 0This process can be repeated for grooming, play and contact networks.
play_spmonkey <- spmonkey_el[spmonkey_el$Beh == "play",][,-c(2,4)] #making a separate data frame for play behavior
sp_play_net_unweighted <- as.network(play_spmonkey, matrix.type = "edgelist", directed = TRUE, weighted=FALSE)
plot(sp_play_net_unweighted, label = "vertex.names", main="Spider Monkey Play Network")
# Adding network and edge attributes to network
sp_play_net_unweighted %v% "sex" <- sp_attr$sex #adds sex as node attribute
sp_play_net_unweighted %v% "age" <- sp_attr$age #adds age as node attribute
sp_play_net_unweighted %v% "repr" <- sp_attr$Repr_status #adds reproductive status as node attributegroom_spmonkey <- spmonkey_el[spmonkey_el$Beh == "groom",][,-c(2,4)] #making a separate data frame for grooming behavior
sp_groom_net <- as.network(groom_spmonkey, matrix.type = "edgelist", directed = TRUE, loops = TRUE)
plot(sp_groom_net, label = "vertex.names", main="Spider Monkey Grooming Network")
# Adding network and edge attributes to the network
sp_groom_net %v% "sex" <- sp_attr$sex #adds sex as node attribute
sp_groom_net %v% "age" <- sp_attr$age #adds age as node attribute
sp_groom_net %v% "repr" <- sp_attr$Repr_status #adds reproductive status as a node attribute
# Let's add the "Mins" from the initial spmonkey edgelist as edge weights
sp_groom_net %e% "time" <- spmonkey_el[spmonkey_el$Beh == "groom",]$Mins contact_spmonkey <- spmonkey_el[spmonkey_el$Beh == "In contact",][,-c(2,4)] #making a separate data frame for contact
sp_contact_net <- as.network(contact_spmonkey, matrix.type = "edgelist", directed = TRUE, loops = TRUE)
plot(sp_contact_net, label = "vertex.names", main="Spider Monkey Contact Network")
# Adding network and edge attributes to the network
sp_contact_net %v% "sex" <- sp_attr$sex #adds sex as node attribute
sp_contact_net %v% "age" <- sp_attr$age #adds age as node attribute
sp_contact_net %v% "repr" <- sp_attr$Repr_status #adds reproductive status as a node attribute
# Let's add the "Mins" from the initial spmonkey edgelist as edge weights
sp_contact_net %e% "time" <- spmonkey_el[spmonkey_el$Beh == "In contact",]$Mins How was this process different from the bonobo data loading, cleaning, and visualizing process? How do these plots compare to the bonobo data? Do you think we could make any predictions based on just looking at these plots?
Now that we have loaded, cleaned, and visualized our spider monkey dataset, we can analyze it via QAP regressions!
QAPs for Spider Monkey Data
What are some questions you would ask about our spider monkey social networks and which QAP would you use?
Correlation
We need to use two networks with the same number of nodes so we chose to ask a question about grooming and contact.
Does grooming relationships correlate with contact relationships in spider monkeys?
What is our null hypothesis? Alternative hypothesis?
# get the correlation value
gcor(sp_groom_net, sp_contact_net)## [1] 0.362977# is it significant?
qap_cor <- qaptest(list(sp_groom_net, sp_contact_net), # include both network objects in a list
gcor, # the function you're using is correlation between networks (gcor)
g1=1, # use first graph in the list
g2=2, # use second graph in the list
reps = 1000) # number of permutations to run
summary(qap_cor)##
## QAP Test Results
##
## Estimated p-values:
## p(f(perm) >= f(d)): 0.003
## p(f(perm) <= f(d)): 0.998
##
## Test Diagnostics:
## Test Value (f(d)): 0.362977
## Replications: 1000
## Distribution Summary:
## Min: -0.3266793
## 1stQ: -0.09074425
## Med: 0
## Mean: 0.002577137
## 3rdQ: 0.09074425
## Max: 0.4355724plot(qap_cor)
How do you interpret this? Are these highly correlated networks? Why is the upper tailed test significant and not the lower tailed test?
Linear regression
Recall that in linear regression for social network analysis involves identifying an independent variable (node attributes) to test its effect on the edge weight of the response variable (the social network). For the spider monkeys, we have multiple options for both node attributes and social networks. We chose to look at age and time spent grooming. Before we can get into the analysis, we must first make the matrices for senders and receivers of grooming like we did for the bonobos.
Is age category predictive of time spent in grooming behavior within dyads of spider monkeys?
NOTE: Senders are represented by ROWS in the matrices
nodes <- length(sp_attr$ID) # the number of nodes in the data set
age <- sp_groom_net %v% "age" # a vector of the node-level variable we are interested in
age_sending <- matrix(data = NA, nrow = nodes, ncol = nodes) #create an empty matrix to be filled
for (i in 1:nodes){ # for 1 through the number of nodes in the data set
age_sending[i,] <- rep(age[i], nodes)} # The age of each actor is repeated over entire ROW of matrix
head(age_sending)## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]
## [1,] 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## [2,] 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## [3,] 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## [4,] 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## [5,] 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## [6,] 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## [,15] [,16] [,17]
## [1,] 3 3 3
## [2,] 3 3 3
## [3,] 3 3 3
## [4,] 3 3 3
## [5,] 3 3 3
## [6,] 3 3 3NOTE: Receivers are represented by COLUMNS in the matrices
age_receiving <- matrix(data = NA, nrow = nodes, ncol = nodes)
for (i in 1:nodes){ # for 1 through the number of nodes in the data set
age_receiving[,i] <- rep(age[i], nodes)} # The age of each actor is repeated over entire COLUMN of matrix
head(age_receiving)## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]
## [1,] 3 3 3 3 3 3 3 3 3 3 3 2 2 2
## [2,] 3 3 3 3 3 3 3 3 3 3 3 2 2 2
## [3,] 3 3 3 3 3 3 3 3 3 3 3 2 2 2
## [4,] 3 3 3 3 3 3 3 3 3 3 3 2 2 2
## [5,] 3 3 3 3 3 3 3 3 3 3 3 2 2 2
## [6,] 3 3 3 3 3 3 3 3 3 3 3 2 2 2
## [,15] [,16] [,17]
## [1,] 1 2 1
## [2,] 1 2 1
## [3,] 1 2 1
## [4,] 1 2 1
## [5,] 1 2 1
## [6,] 1 2 1What is the null hypothesis? What is the alternative hypothesis?
lrqap.sp <- netlm(sp_groom_net, # response variable is a network object with a weighted adjacency matrix
list(age_sending)) # list of all predictor variables as network objects or matrix objects
summary(lrqap.sp)##
## OLS Network Model
##
## Residuals:
## 0% 25% 50% 75% 100%
## -0.3642857 -0.3642857 -0.1214286 0.2500000 0.6357143
##
## Coefficients:
## Estimate Pr(<=b) Pr(>=b) Pr(>=|b|)
## (intercept) -0.3642857 0.048 0.952 0.110
## x1 0.2428571 1.000 0.001 0.008
##
## Residual standard error: 0.4001 on 270 degrees of freedom
## Multiple R-squared: 0.1524 Adjusted R-squared: 0.1492
## F-statistic: 48.54 on 1 and 270 degrees of freedom, p-value: 2.472e-11
##
##
## Test Diagnostics:
##
## Null Hypothesis: qap
## Replications: 1000
## Coefficient Distribution Summary:
##
## (intercept) x1
## Min -6.826e+00 -8.888e+00
## 1stQ -1.974e+00 -1.815e+00
## Median -3.547e-01 2.095e-14
## Mean -2.031e-01 2.470e-02
## 3rdQ 1.422e+00 2.020e+00
## Max 1.045e+01 6.967e+00Does age have a positive or negative relationship with amoung ot time spent grooming? For each unit (in this case life history stage) of age increase, how much more or less likely is a spider monkey to groom? Is the relationship between age and grooming significant? How much variability in the data does this model explain? Is it a good fit model? Is this is a significant effect? How can you tell?
Logistic regression
Recall that for logistic regression we are testing the presence or absence of ties based on the predictor variable (node attribute). Here we chose to look at the effect of sex of individuals on the presence or absence of ties in the play network.
Does sex predict play behavior (number of ties in play social network)? In other words, do males or females play more?
Again, we make the matrices for the variables for both sending and receiving individuals.
NOTE: Senders are represented by ROWS in the matrices
nodes <- 16 # the number of nodes in the data set
sex <- sp_play_net_unweighted %v% "sex" # a vector of the node-level variable we are interested in
sex_sending <- matrix(data = NA, nrow = nodes, ncol = nodes) #create an empty matrix to be filled
for (i in 1:nodes){ # for 1 through the number of nodes in the data set
sex_sending[i,] <- rep(sex[i], nodes)} # age of each actor is repeated over entire ROW of matrix
head(sex_sending)## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]
## [1,] 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## [2,] 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## [3,] 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## [4,] 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## [5,] 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## [6,] 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [,15] [,16]
## [1,] 0 0
## [2,] 0 0
## [3,] 0 0
## [4,] 0 0
## [5,] 0 0
## [6,] 1 1NOTE: Receivers are represented by COLUMNS in the matrices
sex_receiving <- matrix(data = NA, nrow = nodes, ncol = nodes)
for (i in 1:nodes){ # for 1 through the number of nodes in the data set
sex_receiving[,i] <- rep(sex[i], nodes)} # age of each actor is repeated over entire COLUMN of matrix
head(sex_receiving)## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]
## [1,] 0 0 0 0 0 1 1 1 1 1 1 0 0 0
## [2,] 0 0 0 0 0 1 1 1 1 1 1 0 0 0
## [3,] 0 0 0 0 0 1 1 1 1 1 1 0 0 0
## [4,] 0 0 0 0 0 1 1 1 1 1 1 0 0 0
## [5,] 0 0 0 0 0 1 1 1 1 1 1 0 0 0
## [6,] 0 0 0 0 0 1 1 1 1 1 1 0 0 0
## [,15] [,16]
## [1,] 0 0
## [2,] 0 0
## [3,] 0 0
## [4,] 0 0
## [5,] 0 0
## [6,] 0 0What is the null hypothesis? What is the alternative hypothesis?
lrqap <- netlogit(sp_play_net_unweighted, # response variable is a network object with an unweighted adjacency matrix
list(sex_receiving, sex_sending), # list of predictor variables as network or matrix objects
reps = 1000)
summary(lrqap)##
## Network Logit Model
##
## Coefficients:
## Estimate Exp(b) Pr(<=b) Pr(>=b) Pr(>=|b|)
## (intercept) -2.4875920 0.08310985 0.000 1.000 0.000
## x1 0.4783349 1.61338572 0.895 0.105 0.225
## x2 0.1100466 1.11633005 0.559 0.441 0.884
##
## Goodness of Fit Statistics:
##
## Null deviance: 332.7106 on 240 degrees of freedom
## Residual deviance: 150.4112 on 237 degrees of freedom
## Chi-Squared test of fit improvement:
## 182.2994 on 3 degrees of freedom, p-value 0
## AIC: 156.4112 BIC: 166.8531
## Pseudo-R^2 Measures:
## (Dn-Dr)/(Dn-Dr+dfn): 0.4316829
## (Dn-Dr)/Dn: 0.5479219
## Contingency Table (predicted (rows) x actual (cols)):
##
## 0 1
## 0 217 23
## 1 0 0
##
## Total Fraction Correct: 0.9041667
## Fraction Predicted 1s Correct: NaN
## Fraction Predicted 0s Correct: 0.9041667
## False Negative Rate: 1
## False Positive Rate: 0
##
## Test Diagnostics:
##
## Null Hypothesis: qap
## Replications: 1000
## Distribution Summary:
##
## (intercept) x1 x2
## Min -6.830859 -2.051391 -2.690130
## 1stQ -5.254470 -0.663849 -1.139935
## Median -4.616653 0.090777 0.121311
## Mean -4.409391 0.006762 0.035679
## 3rdQ -3.737862 0.604052 1.053686
## Max 0.303001 2.396992 3.798673Do either sex receiving or sex sending have a significant relationship with amount of edges in the play network? Does this model significantly explain the effect?
References
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